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Let $f,g : X \to Y$ be homotopic (quasiconformal) maps between hyperbolic Riemann surfaces $X,Y$. Consider their (unique) lifts $\tilde{f},\tilde{g}: H\to H$ , that fix $0,1,\infty $. My question is : A) what is the maximal subset of the real line $R$, where these two lifts are equal ? Also, conversely,B) if $\tilde{f}=\tilde{g} $on $R$, can we say that $f,g$ are homotopic ? What is the case for the underlying Fuchsian group $\Gamma$ for $X$ when these two lifts are equal on the whole real line $R$ ?

I have been trying to give the proof of A) of the proof of this myself using the concept of limit sets of the Fuchsian group $\Gamma$. To solve this, we could assume,after post-composing with $g$, that $f$ is homotopic to $Id$, without loss of generality. So we need to prove that $\tilde{f}=Id$ on $R$. The also, the converse question becomes: can we say that $f$ is homotopic to $Id$ if $\tilde{f}=Id$ on R ?

Effort of the proof of A):

Since $f$ is homotopic to $Id$, at the level of fundamental group with a fixed base point, $f_*=Id$ if $\tilde{f}$ is $Id$ on $R$.

But on the other hand, at the level of of deck transformation groups, $f_*$ is nothing but conjugation by $f$, am I right, (please correct me if I am not) ? So we have $\tilde{f}\circ \gamma \circ \tilde{f}^{-1}= \gamma \forall \gamma \in \Gamma$, which gives me that $\tilde{f}=Id$ only on the limit set $\Lambda(\Gamma)$ of $\Gamma$ . But when should it happen that $\tilde{f}=Id$ on $R$, if $\Lambda(\Gamma)\ne R$ ?

Also, my related question would be what are all the case of the Fuchsian group $\Gamma$ where the limit set $\Lambda(\Gamma)= R$ ?

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A: Limit set and a priori nothing else. B: Yes, they will be homotopic. C: If tracks of homotopy are uniformly bounded in hyperbolic metric, then extension to the circle is the identity no matter what the limit set is. D ("...what are all the case..."): This question is as meaningful as "what are all the cases when a number is prime". –  Misha Aug 29 '12 at 20:32
    
@ Misha : Thanks for the comment. For part D, I meant can you characterize the Fuchsian groups or the underlying Riemann surfaces $\frac{H}{\Gamma}$ such that $\Lambda(\Gamma)\ne R$. For example, what if I take a Fuchsian group $\Gamma$ so that the underlying surface is of finite analytic type ? surface of infinite analytic type, say of infinite genus ? –  Analysis Now Aug 29 '12 at 22:30
    
Hypgeo: Yes, every surface of finite analytical type has finite hyperbolic area, so the limit set is the full boundary circle of the hyperbolic plane. For surfaces of finite topological type: Limit set is the full circle $\iff$ analytical type is finite. Your next question still makes no sense: Sometimes yes, sometimes not (just like prime numbers). This relates to the concept of "type" of a Riemann surface, see Ahlfors' book "Riemann surfaces" ("parabolic" type $\Rightarrow$ limit set is full circle, but not conversely). –  Misha Aug 30 '12 at 4:40
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